دانلود کتاب Real Analysis

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Cover
Contents
Preface
About the Author
Chapter 1: Basic Properties of the Real Number System
1.1 Introduction
1.2 Order Structure of the Real Number System
1.3 Real Numbers and Decimal Expansions
1.4 The Extended Real Number System
1.5 Complex Field
1.6 The Euclidean Spaces
Solved Exercises
Unsolved Exercises
Chapter 2: Some Finer Aspects of Set Theory
2.1 Introduction
2.2 Russel’s Paradox
2.3 Axiom of Choice
2.4 Sequences, Finite and Infinite Sets
2.5 Countable and Uncountable Sets
2.6 Cantor’s Inequality
2.7 Continuum Hypothesis
Solved Exercises
Unsolved Exercises
Chapter 3: Sequences and Series
3.1 Introduction
3.2 Concepts Connected with Sequences
3.3 Basic Properties of Sequences and Series
3.4 Algebra of Series
3.5 Rearrangement of Series
Solved Exercises
Unsolved Exercises
Chapter 4: Topological Aspects of the Real Line
4.1 Introduction
4.2 The Notion of Distance and the Idea of a Metric Space
4.3 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 5: Limits and Continuity
5.1 Introduction
5.2 Limits
5.3 Continuity
5.4 Discontinuities
5.5 Monotonic Functions
5.6 Uniform Continuity
5.7 Exponents
5.8 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 6: Differentiation
6.1 Introduction
6.2 Definition of Derivative, Examples and Arithmetic Rules
6.2.1 Arithmetic Rules
6.3 Local Extrema and Meanvalue Theorems
6.4 Taylor’s Theorem
6.5 L’Hospital’s Rule
Solved Exercises
Unsolved Exercises
Chapter 7: Functions of Bounded Variation
7.1 Introduction
7.2 Definition and Examples
7.3 Properties of Total Variation
7.4 Functions of Bounded Variation and Monotonic Functions
7.5 Rectifiable Curves
7.6 Absolute Continuity
7.7 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 8: Riemann Integration
8.1 Introduction
8.2 Definition of the Riemann Integral and Examples
8.3 Properties of Riemann Integrals
8.4 Riemann Sums
8.5 Properties of Riemann Integrals
8.6 Meanvalue Theorems for Integral Calculus and the Rule for Change of Variable
8.7 Improper Integrals
8.8 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 9: Sequences and Series of Functions
9.1 Introduction
9.2 Pointwise Convergence, Bounded Convergence and Uniform Convergence
9.3 Properties
9.4 Families of Functions
9.5 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 10: Power Series and Special Functions
10.1 Introduction
10.2 Power Series
10.3 Exponential, Logarithm and Trigonometric Functions
10.4 Beta and Gamma Functions
10.5 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 11: Fourier Series
11.1 Introduction
11.2 Definitions and Examples
Solved Exercises
Unsolved Exercises
Chapter 12: Real-Valued Functions of Two Real Variables
12.1 Introduction
12.2 Limits and Continuity
12.3 Differentiability
12.4 Higher Order Partial Derivatives
12.5 Extreme Values for a Function of Two Variables
12.6 Integration of Functions of Two Real Variables
12.7 Double Integrals
12.8 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 13: Lebesgue Measure Andintegration
13.1 Introduction
13.2 Outer Measure and Measurable Sets
13.2.1 Measurable Sets
13.3 Measurable Functions
13.4 Lebesgue Integral
13.5 Integration of Real-Valued Functions
13.6 Generalizations
Solved Exercises
Unsolved Exercises
Chapter 14: Lp-Spaces
14.1 Introduction
14.2 Definitions and Examples
14.3 Properties of Lp -Spaces
14.4 Fourier Series on L1 [−π, π] and L2 [−π, π]
14.5 Generalizations
Solved Exercises
Unsolved Exercises
Bibliography
Index